Twin blade internal mixers are the main devices used in rubber processing industry to mix natural and synthetic rubbers with carbon black and other compounding ingredients. To facilitate mixing, these internal mixers are always partially filled. Consequently, the flow regime that becomes established inside the mixer is a multiple random free surface flow with moving external boundaries. The stress field associated with this flow regime is usually very uneven. In addition, the flow stream is continuously divided and rejoined by the action of the counter rotating blades. The combination of these complex effects results in the break down, dispersion and distribution of carbon black agglomerates within the rubber matrix. The efficiency of the entire process mainly depends on the geometry of the mixer blades and the operating conditions. Predictive computer modelling offers a very convenient method for the quantitative analysis of mixing process and can be used to design more efficient internal mixers. However, the successful modelling of a complex process such as rubber mixing requires the development and use of sophisticated mathematical algorithms that can take into account its main characteristics. In this respect, a major difficulty is the imposition of the transient boundary conditions in a continuously varying flow domain. In the present paper we describe a robust method which can very effectively resolve this problem. This method is based on the combination of Lagrangian and Eulerian approaches for the modelling of moving boundary flows. We have used this scheme to simulate flow and mixing in the cross-sectional plane of the blades of a tangential rotor internal mixer.